Calculating the Eigenvectors of Diagonally Dominant Matrices

ABSTRACT

An algorithm is proposed for calculating the eigenvectors of a diagonally dominant matrix all of whose elements are known to high relative accuracy. Eigenvectors corresponding to pathologically close eigenvalues are treated by computing the invariant subspace that they span. If the off-diagonal elements of the matrix are sufficiently small, the method is superior to standard techniques, and indeed it may produce a complete set of eigenvectors with an amount of work proportional to the square of the order of the matrix. An analysis is given of the effects of perturbations in the matrix on the eigenvectors.

REFERENCES

• 1: COLLAR, A. R. Some notes on Jahn's method for the improvement of approximate latent roots and vectors of a square matrix. Quart. J. Mech. Appl. Math. 1 (1948) 145-148.
• 2: EBP.R~EI~, P. j. A Jacobi-like method for the automatic computation of eigenvalues and eigenvectors of an arbitrary matrix. SIAM J. 10 (1962), 74-88.
• 3: EBERLEIN, P.J. Solution to the complex eigenproblem by a norm reducing Jacobi type method. Numer. Math. 14 (1970), 232-245.
• 4: EBERLEIN, P. J., AND BOOTHROYD, J. Solution to the eigenproblem by a norm reducing Jacobi type method. Namer. Math. 11 (1968), 1-12.
• 5: HOUSEHOLDER, A.S. The Theory of Matrices in Numerical Analysis. Blaisdell, New York, 1964.
• 6: JAHN, H.A. Improvement of an approximate set of latent roots and modal columns of a matrix by methods akin to those of classical perturbation theory, Quart. J. Mech. Appl. Math. I (1948), 131-144.
• 7: James M. Ortega , Werner C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2000
• 8: RUHE, A. On the quadratic convergence of a generalization of the Jacobi method to arbitrary matrices. BIT 8 (1968), 210-231.
• 9: RUTISHAUSER, H. The Jacobi method for real symmetric matrices. Numer. Malh. 9 (1966), i-i0.
• 10: STEWART, G.W. Error bounds for invariant subspaces of closed operators. SIAM J. Numer. Anal. 8 (1971), 196-808.
• 11: VARGA, R. S. Matrix Iterative Analysis. Prentice-Hzll, Englewood Cliffs, N. J., 1962.
• 12: J. H. Wilkinson, The algebraic eigenvalue problem, Oxford University Press, Inc., New York, NY, 1988
• 13: WILKINSON, j. M., AND REINSCH, C., Eds. Handbook for Automatic Computation, Volume iI, Linear Algebra. Springer-Verlag, Berlin, 1971.

INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations on matrices

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Eigenvalues and eigenvectors (direct and iterative methods)

General Terms:
Algorithms, Theory, Verification